I am trying to find all the eigenvalues of $P$ defined below:
$$P=\begin{bmatrix} 0.5&0.5&0&0&\cdots 0&0 \\ 0.25&0.5&0.25&0&\cdots 0&0\\ 0&0.25&0.5&0.25&\cdots 0&0\\ \vdots \\ 0&0&0&0&\cdots 0.5&0.5 \end{bmatrix}_{n\times n}$$
So $P$ has $0.5$ along the main diagonal. It has $0.25$ on diagonals above and below the main diagonal except for the first and last row. Hence $P$ is not exactly a Toeplitz matrix.
My attempt: A paper I'm looking at, gives eigenvalues of the following Toeplitz matrix
$$ Q=\begin{bmatrix} b&a \\ c&b&a \\ &\ddots&\ddots&\ddots \\ &&c&b&a \\ &&&c&b \end{bmatrix} $$
as $\lambda_j = b+2\sqrt{ca}\cos\left(\frac{j\pi}{n+1} \right)$
So I'm wondering if I will be able to find eigenvalues of $P$ even though its not a Toeplitz matrix precisely?